A wooden block 4mx1mx0.5m is floating in water. Its specific gravity is 0.76. Find the volume of the concrete of specific gravity 2.5, that may be placed on the block which will immerse the (a) block completely in water and (b) block and concrete completely in water.

A wooden block 4mx1mx0.5m is floating in water. Its specific gravity is 0.76. Find the volume of the concrete of specific gravity 2.5, that may be placed on the block which will immerse the

(a) block completely in water and

(b) block and concrete completely in water.

Buoyancy Analysis of a Wooden Block with Concrete

Problem Statement

A wooden block with dimensions:

Length: 4m
Width: 1m
Height: 0.5m
Specific gravity of wood: 0.76
Specific gravity of concrete: 2.5

Determine the volume of concrete required to:

Fully submerge the wooden block in water.
Fully submerge both the wooden block and the concrete in water.

Solution

1. Calculate Volume of Concrete to Fully Submerge the Wooden Block

The total weight of the block and concrete must equal the weight of the displaced water: \[ \gamma_{\text{wood}} V_{\text{wood}} + \gamma_{\text{concrete}} V_{\text{concrete}} = \gamma_{\text{water}} V_{\text{displaced water}} \] Substituting values: \[ 0.76 \times 9810 \times (4 \times 1 \times 0.5) + 2.5 \times 9810 \times V_{\text{concrete}} = 9810 \times (4 \times 1 \times 0.5) \] Solving for \( V_{\text{concrete}} \): \[ V_{\text{concrete}} = 0.192 \text{ m}^3 \]

2. Calculate Volume of Concrete to Fully Submerge Both the Block and Concrete

The total weight of the block and concrete must now equal the weight of the displaced water, including the volume of concrete: \[ \gamma_{\text{wood}} V_{\text{wood}} + \gamma_{\text{concrete}} V_{\text{concrete}} = \gamma_{\text{water}} (V_{\text{wood}} + V_{\text{concrete}}) \] Substituting values: \[ 0.76 \times 9810 \times (4 \times 1 \times 0.5) + 2.5 \times 9810 \times V_{\text{concrete}} = 9810 \times [(4 \times 1 \times 0.5) + V_{\text{concrete}}] \] Solving for \( V_{\text{concrete}} \): \[ V_{\text{concrete}} = 0.32 \text{ m}^3 \]

Final Results:

Volume of concrete required to fully submerge the wooden block: 0.192 m³
Volume of concrete required to fully submerge both the block and concrete: 0.32 m³

Explanation

1. Floating Equilibrium:
The wooden block floats because its density is lower than that of water. To submerge the block completely, additional weight (concrete) must be added.

2. Buoyancy Force and Equilibrium:
The total weight of the block and added concrete must equal the buoyant force exerted by the displaced water. By solving for the volume of concrete required, we find the minimum additional weight needed to achieve equilibrium.

3. Difference Between Partial and Complete Submersion:
– In part (a), only the block needs to be submerged, so less concrete is required.
– In part (b), both the block and concrete need to be submerged, requiring more concrete.

4. Importance of Understanding Buoyancy:
This principle is critical in designing floating structures such as ships, pontoons, and barges. Proper weight distribution ensures stability and prevents capsizing.

Physical Meaning

1. Application in Engineering:
Engineers use similar calculations when designing boats, platforms, and other floating structures to maintain stability and prevent sinking.

2. Load Distribution in Marine Vessels:
Proper load management ensures that a vessel remains stable and does not tip or sink when cargo is added.

3. Real-World Relevance:
This problem illustrates how floating objects behave when additional loads are applied, providing insight into ship design and buoyancy control.